Method and Apparatus of Sharing Spectrum with Legacy Communication System

ABSTRACT

A method of sharing spectrum with a legacy communication system includes acquiring spectrum correlation of the legacy communication system, and generating a transmit waveform based on the spectrum correlation. Unlike conventional cognitive radios that utilize spectrum holes only, the proposed method can also utilize spectrally correlated frequency components where primary-user signals are present.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority of Korean PatentApplication No. 10-2008-0044864 filed on May 15, 2008 which isincorporated by reference in its entirety herein.

FIELD OF THE INVENTION

The invention relates to a method and apparatus for sharing frequencybands with a legacy communication system.

DESCRIPTION OF THE RELATED ART

One of the fundamental problems in communication engineering is todesign a jointly optimal pair of a transmitter and a receiver thatreliably transfers information over a channel between a source and asink. Among various joint transmitter-receiver optimization problems,designing the optimal transmit and receive waveforms for a transmitteremploying linear modulation and an receiver having a linear filterfront-end has long been paid much attention, where the objective is tominimize the mean-squared error (MSE) or, equivalently, maximize thesignal-to-interference-plus-noise ratio (SINR) at the output of thelinear receiver.

Recently, cognitive radio technology has attracted a lot of researchinterest for their potential to dramatically increase intensity andefficiency in radio spectrum utilization. Since next generation wirelesscommunication systems require more frequency resources, it is necessaryto efficiently utilize limited frequency resources. However, in somestudy, it is shown that the spectral efficiency is below 30% inpractical wireless communication systems. The cognitive radio technologyis a method to share unused frequency resources between wirelesscommunication systems. In cognitive radio communication system, asecondary user (also refer to as an unlicensed user) performs spectrumsensing to search unused frequency resources which are not used by aprimary user (also refer to as a licensed user). The secondary user mayutilize the unused frequency resources to improve spectral efficiency.

In the cognitive radio technology, one of design criteria is to minimizethe interference between the primary user and the secondary user.

In W. Wu, S. Vishwanath, and A. Arapostathis, ‘Capacity of a class ofcognitive radio channels-interference channels with degraded messagesets’, IEEE Transactions on Information Theory, vol. 53, no. 11, pp.4391-4399, November 2007, information-theoretic approaches are taken tocognitive radio design with the channel modeled as a two-sendertwo-receiver interference channel. The capacity region is characterizedand/or its bounds are derived, where in general a rate pair isachievable through joint code design at the primary and the secondarytransmitters. The joint code design necessitates the update orreplacement of a single-user encoder/decoder pair in the primary-userside to allow secondary-user service. Such an update may not bedesirable in some applications of the cognitive radio technology.

In A. Jovicic and P. Viswanath, ‘Cognitive radio: aninformation-theoretic perspective’, in proceedings IEEE InternationalSymposium on Information Theory (ISIT)'06, Seattle, USA, Jul. 9-14,2006, pp. 2413-2417, a similar approach is taken under the constraintthat no interference is generated to the primary user, where nointerference means that no penalty in primary user's SINR is incurred bythe secondary transmitter. This cognitive radio is assumed to have anaccess to the message of the primary transmitter. This enables thecognitive radio to transmit its own data stream and, at the same time,to transmit the data stream of the primary user, so that the power ofthe primary user signal as well as that of the interference can beincreased simultaneously at the primary receiver. Thus, a fixedsingle-user transmitter-receiver pair is used at the primary side whilethe secondary user enters and exits the frequency band. Suchunidirectional cooperation of the secondary transmitter, however, is notpossible if no message of the primary transmitter is available to thesecondary transmitter. Of course, a cognitive radio must be provideddirectly or indirectly some information about the primary user'ssignaling and system parameters in order to tactically design itstransmitted signal. However, the assumption of known message may beexcessive for the practical implementation of a cognitive radio.

In L. Zhang, Y.-C. Liang, and Y. Xin, ‘Joint beamforming and powerallocation for multiple access channels in cognitive radio networks’,IEEE Journal on Selected Area in Communication, vol. 26, no. 1, pp.38-51, January 2008, cognitive radios are designed that induce nonzerobut limited amount of interference at the primary receivers without theassumption of known message. The interference is simply defined as thetransmit power of the secondary user times the channel gain from thesecondary transmitter to the primary receiver. So, the amount ofinformation needed at the secondary transmitter is very small. Thispre-processing interference, however, may not be relevant enough toproperly assess the performance degradation of the primary users thanthe post-processing interference that can be defined as the amount ofinterference obtained after the primary receivers perform front-endand/or following signal processing.

SUMMARY OF THE INVENTION

A method and apparatus for sharing spectrum with at least one legacycommunication system is provided.

In an aspect, a method of sharing spectrum with a legacy communicationsystem includes acquiring spectrum correlation of the legacycommunication system, and generating a transmit waveform based on thespectrum correlation.

The method may further include generating a transmit signal by combiningthe transmit waveform and a data signal, and transmitting the transmitsignal.

In an embodiment, the spectrum correlation may be related to statisticalcharacteristics of the legacy communication system or be related tocorrelations of signals of the legacy communication system in frequencydomain. The transmit waveform may be generated by using VectorizedFourier Transform (VFT).

In an embodiment, the transmit waveform may be generated by determininga power spectral density (PSD) matrix R_(N)(ƒ) based on the spectrumcorrelation, determining a channel correlation matrix C(ƒ) by using thePSD matrix R_(N)(ƒ) and a channel matrix H(ƒ), determining a projectionmatrix P(ƒ), determining an eigenvalue λ_(max)(ƒ) of P(ƒ)C(ƒ),determining a normalized eigenvector v_(max)(ƒ) corresponding to theλ_(max)(ƒ), and determining the transmit waveform s_(opt)(ƒ) from theeigenvector v_(max)(ƒ). The eigenvalue λ_(max)(ƒ) may be the largesteigenvalue of the P(ƒ)C(ƒ). The projection matrix P(ƒ) may be determinedby using a blocking matrix G(ƒ) obtained from impulse responses of thelegacy communication system.

In another aspect, a transmitter for sharing spectrum with a legacycommunication system includes a waveform generator for generating atransmit waveform, and a Radio Frequency (RF) unit for transmitting aradio signal based on the transmit waveform. The waveform generator isconfigured to acquire spectrum correlation of the legacy communicationsystem, and generate the transmit waveform based on the spectrumcorrelation.

In still another aspect, a receiver for sharing spectrum with a legacycommunication system includes a waveform generator for generating areceive waveform, and a RF filter for filtering a receive signal basedon the receive waveform. The waveform generator is configured to acquirespectrum correlation of the legacy communication system, and generatethe receive waveform based on the spectrum correlation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a typical cognitive radio channel.

FIG. 2 shows overlay and legacy wireless communication systems.

FIG. 3 shows a block diagram in complex baseband.

FIG. 4 shows an example of transmit band, receive band, and frequencyresponse.

FIG. 5 shows PSDs of the signals of the overlay and legacy systems whenroll-off factor equals 0.5.

FIG. 6 shows average BER versus normalized hole length.

FIG. 7 shows trade-off between the average BER and the symbol rate whenroll-off factor equals 0.22, and 1/T=1/(4T₀).

FIG. 8 shows average BER versus roll-off factor of the legacy system'ssquare-root raised cosine (SRRC) transmitter waveform.

FIG. 9 shows comparison of the proposed scheme that utilizes spectrumhole only with the proposed scheme.

FIG. 10 shows average BER versus Eb/No for various transmit bands.

FIG. 11 shows a method of sharing spectrum with a legacy communicationsystem according an embodiment of the present invention.

FIG. 12 shows a flowchart of generating a transmit waveform.

FIG. 13 shows a block diagram of a transmitter and a receiver of anoverlay communication system according an embodiment of the presentinvention.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, a legacy communication system means a communication systemwhich uses a licensed frequency band. The legacy communication systemprovides services to a primary user or a licensed user. An overlaycommunication system means a communication system which shares thefrequency band with one or more legacy communication systems. Theoverlay communication system provides services to a secondary user or anunlicensed user. The overlay communication system may also be referredto as a cognitive radio system.

FIG. 1 shows a cognitive radio channel. There are three primary usersignals inside the frequency band of interest, and a white space withoutthe primary user signals consists of four spectrum holes. Since theutilization of the frequency component other than the spectrum holes mayresult in a dramatic increase of the spectral efficiency, we make thefollowing assumptions: 1) the overall primary user signal issecond-order wide-sense cyclostationary (WSCS), 2) each primary receiveris equipped with a linear filter front-end whose output is sampled at aninteger multiple of the cycle frequency, 3) the information about thesecondary-to-primary channels, the receive-filter waveforms, and thesampling timing of the primary receivers is available at the secondarytransmitter. Notice that linear modulation, which is one of the mostpopular modulation schemes for digital communication, results incyclostationarity of the transmitted signal with the fundamental cyclefrequency equal to the symbol rate and is usually accompanied by alinear filter front-end whose output is sampled at an integer multipleof the symbol rate. It will be shown that, under these assumptions andcertain conditions, it is possible to find the optimal pair of thetransmit and the receive waveforms that can utilize the gray and theblack spaces and, at the same time, induces zero interference to theoutputs of the primary receivers.

FIG. 2 shows overlay and legacy wireless communication systems. Legacysystems 100, 110 transmit and receive data using authorized frequencybands. Although an overlay system 130 has no authority to use thefrequency bands, the overlay system 130 shares frequency bands with thelegacy systems 100, 110 when the signals of the overlay system 130 givesno interference the signals of the legacy systems 100, 110. To minimizeinterference, the overlay system 130 uses a transmit waveform based onspectrum correlation of the legacy systems 100, 110. The overlay system130 may acquire the spectrum correlation from the legacy systems 100,110 via signaling or from sensing the spectrum of the legacy systems100, 110.

The overlay system 130 includes a transmitter 131 and a receiver 132. Indownlink transmission, the transmitter 131 may be a part of a basestation and the receiver 132 may be a part of a user terminal. In uplinktransmission, the transmitter 131 may be a part of a user terminal andthe receiver 132 may be a part of a base station. The transmitter 132transmits signals to the receiver 132 in the frequency band occupied ornot used by the legacy systems 100, 110.

I. System Model and Problem Formulation

In this section, we describe the system model and formulate theoptimization problem in the time domain.

A. System Model

The system block diagram in complex baseband is depicted in FIG. 3.There are K primary users operating over the frequency band of interest.It is assumed that the overall primary-user signal is zero-meanproper-complex WSCS with fundamental cycle period T₀ [sec]. It is alsoassumed that the kth primary Rx employs a linear filter front-end withimpulse response w_(k)(−t)*, for k ∈

where the superscript * denotes complex conjugation and

{1,2, . . . ,K} denotes the index set of the primary users. The sequenceof decision statistics (z_(k)[l])_(l) at the kth primary Rx, for

{1,2, . . . ,K}, is obtained by sampling the filter output at everyT₀/M_(k) [sec] with offset τ_k ∈ [0,T₀/M_(k)), where M_(k) is a positiveinteger. For example, suppose that there are two primary users employinglinear modulation with symbol periods T₁ and T₂ (=2T₁/3), respectively,and that their linear filter outputs are oversampled by a factor of two.Then, we have T₀=2T₁=3T₂, M₁=4, and M₂=6.

A secondary user that employs linear modulation with symbol transmissionrate 1/T [symbols/sec] is to be designed. It is assumed that the symbolperiod T(=M₀T₀) is an integer multiple of T₀. The impulse response ofthe transmit filter is denoted by s(t), and that of the receive filteris denoted by w(−t)*. Thus, the input to the channel is Σ_(l=−∞)^(∞)b[l]δ(t−lT)

s(t)=Σ_(l=−∞) ^(∞)b[l]s(t−lT), where δ(t) denotes the Dirac deltafunction and

denotes convolution. The channel from the secondary transmitter (Tx) tothe secondary receiver (Rx) is assumed to be very slowly time-varyingrelatively to the data transmission rate. So, it is modeled as a lineartime-invariant filter with impulse response h(t). The data sequence(b[l])_(l) of the secondary user is assumed to be a proper-complexzero-mean wide-sense stationary (WSS) colored random process withauto-correlation function

m[l]

{b[l+l′]b[l′]*}  (1)

and power spectral density (PSD)

$\begin{matrix}{{{M(f)}\overset{\bigtriangleup}{=}{\sum\limits_{l = {- \infty}}^{\infty}{{m\lbrack l\rbrack}^{{- j}\; 2\; \pi \; {fl}}}}},} & (2)\end{matrix}$

where

{•} denotes expectation. It is also assumed that the secondary-usersignal is uncorrelated with the primary-user signals.

The input z(t) to the secondary Rx is the signal from the secondary Txplus the interference n_(I)(t) from the K primary users, corrupted by anambient noise n_(A)(t). The ambient noise is modeled as a proper-complexadditive white Gaussian noise (AWGN) with two-sided PSD N₀. Thus, z(t)can be written as

$\begin{matrix}{{{z(t)} = {{\sum\limits_{l = {- \infty}}^{\infty}{{b\lbrack l\rbrack}{p\left( {t - {lT}} \right)}}} + {n(t)}}},} & (3)\end{matrix}$

where p(t)

h(t)

s(t) is the response of the channel to the transmit waveform s(t), andn(t) is the overall interference-plus-noise process given by

n(t)

n_(I)(t)+n_(A)(t).   (4)

The sequence of decision statistics (z[l])_(l) is, then, obtained as

z[l]

z(t)

w(−t)*|_(t=lT)   (5)

by sampling the receive filter output at every T [sec]. Without loss ofgenerality, we set the sampling offset at the secondary receiver tozero.

B. Problem Formulation in Time Domain

The objective is to find the secondary user's transmit and receivewaveforms s(t) and w(t) that jointly minimize the MSE

ε

{|b[l]−z[l]|²}  (6)

at the output sequence (z[l])_(l), given a frequency band over which thesecondary user can transmit the signal. Note that the MSE (6) isuniquely defined regardless of the value of l. This is because both theprimary-user and the secondary-user signals are WSCS with the commoncycle period of T.

There are two major constraints imposed on this optimization problem.The first constraint is on the average transmit power of the secondaryuser. Due to the cyclostationarity of the transmitted signal with cycleperiod T, the average transmit power P can be defined as

$\begin{matrix}{{\overset{\_}{P}\overset{\bigtriangleup}{=}{\left\{ {\frac{1}{T}{\int_{\langle T\rangle}^{\;}{{{\sum\limits_{l = {- \infty}}^{\infty}{{b\lbrack l\rbrack}{s\left( {t - {lT}} \right)}}}}^{2}\ {t}}}} \right\}}},} & (7)\end{matrix}$

where

T

denotes any integration interval of length T. Thus, the constraint canbe written as P=A for some A>0.

The second constraint is on the amount of interference induced to theprimary receivers. As mentioned already, we adopt the zero interferenceconstraint at the output samples of the linear filter front-ends of theprimary users. The channel from the secondary Tx to the kth primary Rx,for k ∈

, is modeled as a linear time-invariant filter with impulse responseh_(k)(t). Then, for each k ∈

, the decision statistic z_(k)[l] at the kth primary Rx has theinterference component {circumflex over (z)}_(k)[l] from the secondaryuser, given by

$\begin{matrix}{{{{\hat{z}}_{k}\lbrack l\rbrack}\overset{\bigtriangleup}{=}{\sum\limits_{l^{\prime} = {- \infty}}^{\infty}{{b\left\lbrack l^{\prime} \right\rbrack}{q_{k}\left( {\frac{lT}{M_{0}M_{k}} - {l^{\prime}T}} \right)}}}},} & (8)\end{matrix}$

where the waveform q_(k)(t) is defined as

$\begin{matrix}{{q_{k}(t)}{w_{k}\left( {{- t} - \tau_{k}} \right)}^{*}\left( {{s(t)}{h_{k}(t)}} \right)} & \left( {9a} \right) \\{\mspace{50mu} {{w_{k}\left( {{- t} - \tau_{k}} \right)}^{*}{p_{k}(t)}}} & \left( {9b} \right)\end{matrix}$

to absorb the effect of the sampling offset τ_(k). Thus, the constraintcan be written as {circumflex over (z)}_(k)[l]=0, ∀ k ∈

and ∀ l ∈

, where

denotes the set of all integers.

In summary, the optimization problem is given by

Problem 1:

$\begin{matrix}{\underset{{s{(t)}},{w{(t)}}}{minimize}\left\{ {{{b\lbrack l\rbrack} - {z\lbrack l\rbrack}}}^{2} \right\}} & \left( {10a} \right) \\{{{{subject}\mspace{14mu} {to}\mspace{14mu} \left\{ {\frac{1}{T}{\int_{\langle T\rangle}{{{\sum\limits_{l = {- \infty}}^{\infty}{{b\lbrack l\rbrack}{s\left( {t - {lT}} \right)}}}}^{2}\ {t}}}} \right\}} = A},} & \left( {10b} \right) \\{{{\sum\limits_{l^{\prime} = {- \infty}}^{\infty}{{b\left\lbrack l^{\prime} \right\rbrack}{q_{k}\left( {\frac{lT}{M_{0}M_{k}} - {l^{\prime}T}} \right)}}} = 0},{\forall{k \in}},{\forall{l \in}},} & \left( {10c} \right)\end{matrix}$

which is to be solved under the assumption of perfect knowledge on theimpulse responses h(t), (h_(k)(t))_(k′) and (w_(k)(t))_(k) of thechannels and the receive filters. Of course, there are additionalconstraints, which must be described in the frequency domain. These arethe piecewise smoothness constraint on the Fourier transforms of s(t)and w(t), and the bandwidth constraint. The piecewise smoothnessconstraint is imposed to enable the Fourier transform and to eliminatepathological functions. The bandwidth constraint is discussed in detailin the next section.

II. Problem Reformulation

In this section, Problem 1 described in the time domain is reformulated.Using the VFT technique, the objective function (10a) and theconstraints (10b) and (10c) are rewritten in the frequency domain. Toallow non-identical frequency bands for the signal transmission andreception of the secondary user, the notions of transmit band, receiveband, and virtual primary Rx are also introduced.

A. Conversion to Frequency Domain Problem

The system employs a linear modulation and the received signal iscorrupted by an additive WSCS noise, where the fundamental cyclefrequency is equal to the symbol transmission rate of the system. Thetransmit and the receive waveforms are designed to minimize the MSE atthe output of a linear receiver, subject to the average transmit powerconstraint. Not the ordinary Fourier transform but a specialtransformation technique called the VFT is used to handle the additiveWSCS noise. This VFT technique plays a key role to reformulate and solveProblem 1. To proceed, we need the following definitions.

Definition 1: Given a communication system with bandwidth B [Hz] incomplex baseband and data symbol transmission rate 1/T [symbols/sec],the excess bandwidth β is defined by the relation

$\begin{matrix}{{BT} = {\frac{\left( {1 + \beta} \right)}{2}.}} & (11)\end{matrix}$

If a linear modulation is employed in the system, then the excessbandwidth is the ratio of the extra amount of bandwidth used in thesignaling to the Nyquist minimum bandwidth for zero intersymbolinterference. As shown below, this parameter is closely related to thesignal dimension in the frequency domain.

Definition 2: Let x(t) be a time function with Fourier transform X(ξ)

∫_(−∞) ^(∞)x(t)e^(−j2πξt)dt and bandwidth B [Hz] in complex baseband.Then, its VFT x(ƒ) with sampling rate T [samples/Hz] is defined as

$\begin{matrix}{{x(f)}\overset{\bigtriangleup}{=}\begin{bmatrix}{X\left( {{- \frac{L}{T}} + f} \right)} \\{X\left( {{- \frac{L - 1}{T}} + f} \right)} \\\vdots \\{X\left( {\frac{L}{T} + f} \right)}\end{bmatrix}} & (12)\end{matrix}$

ƒ ∈

, where the integer L is defined as L

┌β/2┐=┌(2BT−1)/2┐ and the Nyquist interval

is defined as

$\begin{matrix}{\overset{\bigtriangleup}{=}{\left\{ {{{f\text{:}} - \frac{1}{2T}} \leq f < \frac{1}{2T}} \right\}.}} & (13)\end{matrix}$

Some of the entries in x(ƒ) at specific values of ƒ are always zero dueto the bandwidth limitation of x(t) to B [Hz], which requires thefollowing refinement on the definition.

Definition 3:

The effective VFT of x(t) is defined as a variable-dimensional functionof ƒ, obtained by removing the first entry of x(ƒ) for

$\begin{matrix}{{- \frac{1}{2T}} \leq f < {{- \frac{1 + \beta}{2T}} + \frac{L}{T}}} & \left( {14a} \right)\end{matrix}$

and the last entry of x(ƒ) for

$\begin{matrix}{{\frac{1 + \beta}{2T} - \frac{L}{T}} \leq f < {\frac{1}{2T}.}} & \left( {14b} \right)\end{matrix}$

For convenience, we define the degree of freedom as follows.

Definition 4: The degree of freedom N(ƒ) as a function of ƒ ∈

is defined as the dimension of the effective VFT at offset ƒ.

The degree of freedom N(ƒ), thus, means the number of free variables inthe signal design at the frequency offset ƒ. In what follows, all VFTsare effective ones. It can be shown that the rules (14a) and (14b) leadto

$\begin{matrix}{{N(f)} = \left\{ \begin{matrix}{{1 + \left\lceil \beta \right\rceil},} & {{{for}\mspace{14mu} {f}} < \frac{1 + \beta - \left\lceil \beta \right\rceil}{2T}} \\{\left\lceil \beta \right\rceil,} & {otherwise}\end{matrix} \right.} & \left( {15a} \right)\end{matrix}$

when ┌β┐ is an even number, and

$\begin{matrix}{{N(f)} = \left\{ \begin{matrix}\left\lceil \beta \right\rceil & {{{for}\mspace{14mu} {f}} < \frac{\beta - \left\lceil \beta \right\rceil}{2T}} \\{{1 + \left\lceil \beta \right\rceil},} & {otherwise}\end{matrix} \right.} & \left( {15b} \right)\end{matrix}$

when ┌β┐ is an odd number. Note that the parameter β determines thesignal dimension at each ƒ ∈

.

Now, we are ready to convert Problem 1 to an equivalent problemdescribed in the frequency domain. First, we convert the objectivefunction (10a). For this, the MSE at the output of the secondary Rx isrewritten as

ε=

{|b[l]| ²}−2

{b[l]*z[l]})+

{|z[l]−n[l]| ² }+

{|n[l]| ²},   (16)

where the noise component n[l] is defined as

$\begin{matrix}{{n\lbrack l\rbrack} = {\int_{- \infty}^{\infty}{{w\left( {t - {lT}} \right)}^{*}{n(f)}\ {{t}.}}}} & (17)\end{matrix}$

Each term in (16) can be rewritten as

$\begin{matrix}{{{\left\{ {{b\lbrack l\rbrack}}^{2} \right\}} = {{{TM}({fT})}\ {f}}},} & \left( {18a} \right) \\{{{{- 2}\left( {\left\{ {{b\lbrack l\rbrack}^{*}{z\lbrack l\rbrack}} \right\}} \right)} = {{- 2}\left( {{w(f)}^{H}\left( {{M({fT})}{p(f)}} \right)} \right){f}}},} & \left( {18b} \right) \\{{{\left\{ {{{z\lbrack l\rbrack} - {n\lbrack l\rbrack}}}^{2} \right\}} = {{w(f)}^{H}\left( {\frac{M({fT})}{T}{p(f)}{p(f)}^{H}} \right){w(f)}{f}}},{and}} & \left( {18c} \right) \\{{{\left\{ {{n\lbrack l\rbrack}}^{2} \right\}} = {{w(f)}^{H}{R_{N}(f)}{w(f)}\ {f}}},} & \left( {18d} \right)\end{matrix}$

respectively, where w(ƒ), p(ƒ) and R_(N)(ƒ) are the VFT of the receivewaveform w(t), the VFT of the channel response p(t) of the transmitwaveform, and the matrix-valued PSD of the WSCS noise n(t),respectively. The (k,l)th entry of the matrix-valued PSD R_(N)(ƒ) can beobtained as

$\begin{matrix}{\left\lbrack {R_{N}(f)} \right\rbrack_{k,l}\left\{ \begin{matrix}{{R_{N}^{({k - l})}\left( {f + \frac{l - L - 1}{T}} \right)},} & {{{for}\mspace{14mu} {{k - l}}} \leq {2L}} \\{0,} & {elsewhere}\end{matrix} \right.} & (19)\end{matrix}$

where R_(N) ^((k))(ξ) and the double Fourier transform R_(N)(ξ,{tildeover (ξ)}) of

{N(t)N(s)*} satisfy the relation

$\begin{matrix}{{R_{N}\left( {\xi,\overset{\sim}{\xi}} \right)} = {\sum\limits_{k}{{R_{N}^{(k)}\left( {\xi - \frac{k}{T}} \right)}{{\delta \left( {\xi - \xi - \frac{k}{T}} \right)}.}}}} & (20)\end{matrix}$

Note that, due to the ambient noise component in n(t), the matrixR_(N)(ƒ) is positive definite ∀ ƒ ∈

. If we further define R_(Z)(ƒ) as the matrix-valued PSD of the receivedsignal z(t), then it can be shown that R_(Z)(ƒ) is given by

$\begin{matrix}{{{R_{Z}(f)}{R_{N}(f)}} + {\frac{1}{T}{M({fT})}{p(f)}{p(f)}^{H}}} & (21)\end{matrix}$

and, consequently, the MSE (6) can be simplified as

$\begin{matrix}{ɛ = {{{{TM}({fT})}} + {{w(f)}^{H}{R_{Z}(f)}{w(f)}} - {2\left( {{w(f)}^{H}\left( {{M({fT})}{p(f)}} \right)} \right)\ {f}}}} & (22)\end{matrix}$

Second, we convert the average power constraint (10b) as

$\begin{matrix}{{\frac{M({fT})}{T}{s(f)}^{H}{s(f)}{f}} = A} & (23)\end{matrix}$

Third, we convert the zero interference constraint (10c) to anorthogonality constraint, as shown in the following lemma. For this, lete_(k,l)(ƒ) for k ∈

and l ∈ {0,1, . . . ,M₀M_(k)−1} be a variable-length vector-valuedfunction of ƒ ∈

defined as

$\begin{matrix}{{e_{k,l}(f)}{\begin{bmatrix}^{j\; 2\; {{\pi {({\frac{l}{M_{o}M_{k}} + \frac{\tau_{k}}{T}})}} \cdot 1}} \\^{j\; 2\; {{\pi {({\frac{l}{M_{o}M_{k}} + \frac{\tau_{k}}{T}})}} \cdot 2}} \\\vdots \\^{j\; 2\; {{\pi {({\frac{l}{M_{o}M_{k}} + \frac{\tau_{k}}{T}})}} \cdot {N{(f)}}}}\end{bmatrix}}} & (24)\end{matrix}$

and let ⊙ denote the entry-by-entry Hadamard product.

Lemma 1: The zero interference constraint (10c) can be expressed as anorthogonality constraint

$\begin{matrix}{{{\sqrt{m({fT})}{w_{k,l}(f)}^{H}{H_{k}(f)}{s(f)}} = 0},{\forall{f \in}},{\forall{k \in}},{\forall{l \in \left\{ {0,1,\ldots \mspace{14mu},{{M_{0}M_{k}} - 1}} \right\}}}} & (25)\end{matrix}$

on s(ƒ), where w_(k,l)(ƒ) is defined as

w_(k,l)(ƒ)

w_(k)(ƒ) ⊙ e_(k,l)(ƒ)   (26)

with w_(k)(ƒ) being the VFT of w_(k)(t),

H_(k)(ƒ)

diag(h_(k)(ƒ))   (27)

is the N(ƒ)-by-N(ƒ) diagonal matrix with h_(k)(ƒ) being the VFT ofk_(k)(t), and s(ƒ) is the VFT of s(t).

Finally, Problem 1 described in the time domain can be transformed to

Problem 2:

$\begin{matrix}{{{\underset{{s{(f)}},{w{(f)}}}{minimize}{{TM}({fT})}} + {{w(f)}^{H}{R_{Z}(f)}{w(f)}} - {2\ \left( {{w(f)}^{H}\left( {{M({fT})}{p(f)}} \right)} \right){f}}}\mspace{85mu} {{subject}\mspace{14mu} {to}}} & \left( {28a} \right) \\{\mspace{79mu} {{{\sqrt{M({fT})}{w_{k,l}(f)}^{H}{H_{k}(f)}{s(f)}} = 0},\mspace{79mu} {\forall{f \in}},{\forall{k \in}},\mspace{79mu} {\forall{l \in \left\{ {0,1,\ldots \mspace{14mu},{{M_{0}M_{k}} - 1}} \right\}}},}} & \left( {28b} \right) \\{\mspace{76mu} {{{\frac{M({fT})}{T}{s(f)}^{H}{s(f)}\ {f}} = A},}} & \left( {28c} \right)\end{matrix}$

which is now described in the frequency domain. At a first glance, thenext step is to directly tackle Problem 2 to find an optimal solution,after checking the existence of a solution. However, there is a subtletythat needs to be discussed, due to the possible difference in bandwidthsof the transmit waveform s(t) and the receive waveform w(t). This issueis dealt with in detail in the next subsection.

B. Transmit Band, Receive Band, and Virtual Primary Receivers

In order to use the VFT technique in the optimization, the centerfrequency used in the complex baseband down-conversion and the bandwidthB used in the vectorization (12) need to be determined first. Toproceed, we introduce the following two definitions.

Definition 5: The transmit band

{ξ: W_(T,0)≦|ξ|≦W_(T,1)} is defined as a frequency band where thesecondary Tx can transmit a signal.

Definition 6: The receive band

{ξ: W_(R,0)≦|ξ|≦W_(R,1)} is defined as a frequency band where thesecondary Rx can receive and process the signal.

Then, there are three cases worth considering in the cognitive radiodesign: 1) Both

and

are given, 2) only

is given, and 3) only

is given. Note that the optimal Tx does not allocate any signal power inthe complement

of

because the signal component in

is not observed by the secondary Rx and, consequently, it cannotcontribute to reducing the MSE. Thus, without loss of optimality, were-define the transmit band as

in case 1), and we set

=

in case 2). In case 3), in order not to waste any transmit power,

must be chosen to include

. Moreover, it is potentially suboptimal not to include a frequency bandhaving an observable that has non-zero correlation with the observableinside

. This is because the non-zero correlation can be exploited in reducingthe MSE. Thus, we set W_(R,0)(≦W_(T,0)) and W_(R,1)(≧W_(T,1)),respectively, as the maximum and the minimum values of frequencies suchthat the observable inside

has no correlated observable outside

, where W_(R,0) and W_(R,1) are assumed finite. In all three cases, thetransmit band is now contained in the receive band. So, it is convenientto choose the center frequency of

as the reference for complex baseband representation and the basebandbandwidth of

as the bandwidth B for the vectorization.

If the transmit band is a proper subset of the receive band, thenconstraints additional to the zero interference constraint (28b) must beimposed on s(t), otherwise the secondary user may emit signal power in

. To handle this situation, we introduce the notion of a virtual primaryRx, where and in what follows

and

are the transmit and the receive bands in complex baseband,respectively, and

is the support of the channel frequency response H(ξ) between thesecondary Tx and the secondary Rx, i.e.,

{ξ: H(ξ)≠0}.   (29)

Definition 7: The lth virtual primary Rx, for l=−L,−L+1, . . . ,L, isdefined as a fictitious primary Rx that has the linear filter front-endwith frequency response

$\begin{matrix}{{{\hat{W}}_{1}(\xi)} = \left\{ \begin{matrix}{{1_{B_{R}{{\backslash(}{{B_{T}\bigcap B_{H}})}}}(\xi)},} & {{{for}\mspace{14mu} \frac{{2l} - 1}{2T}} \leq \xi < \frac{{2l} + 1}{2T}} \\{0,} & {{elsewhere},}\end{matrix} \right.} & (30)\end{matrix}$

and sampling rate 1/T, where 1_({•})(ξ) denotes the indicator function.

As we do not allow any secondary-user signal component existing at thesampled output of the primary receivers, the introduction of suchvirtual primary receivers makes the secondary Tx emit no power outsidethe transmit band. Just to serve this purpose only, the index set

in (30) appears excessive and can be simplified to

. However, we define a virtual primary Rx as Definition 7 because thisdefinition makes the argument much simpler on the necessary andsufficient condition for the existence of the optimal solution, whichwill be discussed in Section IV-B. Note that this modification of theindex set does not result in any loss of optimality, because the optimalsecondary transmitter never emits signal in the frequency band

otherwise the signal component in

is completely nulled out by the channel and, consequently, the energy inthe band is wasted without contributing to minimizing the MSE.

EXAMPLE 1

Suppose that

and H(ξ) are given as illustrated in FIG. 4. Then, there are two virtualprimary receivers with non-zero frequency responses given

${{{by}\mspace{14mu} {{\hat{W}}_{- 1}(\xi)}} = 1},{{{for}\mspace{11mu} - \frac{3}{2T}} \leq \xi < {- \xi_{0}}},{{{\hat{W}}_{1}(\xi)} = 1},{{{for}\mspace{14mu} \xi_{0}} \leq \xi < \frac{3}{2T}},$

and zero, elsewhere. Thus, their VFTs are given by

$\begin{matrix}{{{\hat{w}}_{- 1}(f)} = \left\{ \begin{matrix}{\begin{bmatrix}1 & 0 & 0\end{bmatrix}^{H},} & {{{{for}\mspace{14mu} - \frac{1}{2T}} \leq f < {{- \xi_{0}} + \frac{2}{2T}}},} \\{\begin{bmatrix}0 & 0 & 0\end{bmatrix}^{H},} & {{{{for}\mspace{14mu} - \xi_{0} + \frac{2}{2T}} \leq f < \frac{1}{2T}},}\end{matrix} \right.} & \left( {31a} \right) \\{{{\hat{w}}_{1}(f)} = \left\{ \begin{matrix}{\begin{bmatrix}0 & 0 & 0\end{bmatrix}^{H},} & {{{{for}\mspace{14mu} - \frac{1}{2T}} \leq f < {\xi_{0} - \frac{2}{2T}}},} \\{\begin{bmatrix}0 & 0 & 1\end{bmatrix}^{H},} & {{{{for}\mspace{14mu} - \xi_{0} - \frac{2}{2T}} \leq f < \frac{1}{2T}},}\end{matrix} \right.} & \left( {31b} \right)\end{matrix}$

In contrast to the fact that the optimal S(ξ) cannot allocate energyoutside the frequency band

, the VFT s(ƒ) of the optimal solution can be chosen arbitrarily outsidethe support

of M(ƒT) defined as

{ƒ ∈

: M(ƒT)≠0}  (32)

This is because s(ƒ) is always accompanied by the factor √{square rootover (M(ƒT))} and, consequently, no signal component of s(ƒ) in ƒ ∈

is transmitted by the secondary user. Thus, the range of ƒ in thezero-interference and the transmit power constraints can be confined to

without loss of optimality.

In summary, Problem 2 can be rewritten as

Problem 3:

$\begin{matrix}{{\underset{{s{(f)}},{w{(f)}}}{minimize}\mspace{14mu} {{TM}({fT})}} + {{w(f)}^{H}{R_{Z}(f)}{w(f)}} - {2\left( {{w(f)}^{H}\left( {{M({fT})}{p(f)}} \right)} \right)\ {f}}} & \left( {33a} \right) \\{\mspace{79mu} {{{{subject}\mspace{14mu} {to}\mspace{14mu} {G(f)}^{H}{s(f)}} = 0},{\forall{f \in}},}} & \left( {33b} \right) \\{\mspace{79mu} {{{\frac{M({fT})}{T}\ {s(f)}^{H}{s(f)}{f}} = A},}} & \left( {33c} \right)\end{matrix}$

where the matrix G(ƒ) is defined as

$\begin{matrix}{{G(f)}^{H}{\begin{bmatrix}{{w_{1,0}(f)}^{H}{H_{1}(f)}} \\{{w_{1,1}(f)}^{H}{H_{1}(f)}} \\\vdots \\{{w_{1,{{M_{0}M_{1}} - 1}}(f)}^{H}{H_{1}(f)}} \\\vdots \\{{w_{K,O}(f)}^{H}{H_{K}(f)}} \\{{w_{K,1}(f)}^{H}{H_{K}(f)}} \\\vdots \\{{w_{K,{{M_{0}M_{1}} - 1}}(f)}^{H}{H_{K}(f)}} \\{{\hat{w}}_{- L}(f)}^{H} \\{{\hat{w}}_{{- L} + 1}(f)}^{H} \\\vdots \\{{\hat{w}}_{L}(f)}^{H}\end{bmatrix}}} & (34)\end{matrix}$

with the VFTs ŵ_(l)(ƒ),l=−L,−L+1, . . . ,L, of the receive waveforms ofthe virtual primary receivers augmenting the orthogonality constraint(28b).

III. Optimal Solution and Its Existence

In this section, the optimal solution to Problem 3 is derived. Due tothe zero interference constraint (33b), it is possible for theconstraint set to be empty, which makes the problem vacuous. So, anatural question is when Problem 3 has an optimal solution. This issueof the existence of the solution is also discussed.

A. Derivation of Optimal Solution

The optimization problem Problem 3 can be rewritten in a doubleminimization form

$\begin{matrix}{\underset{s{(f)}}{minimize}\begin{Bmatrix}{{\underset{w{(f)}}{minimize}{{TM}({fT})}} + {{w(f)}^{H}{R_{Z}(f)}w(f)}\  -} \\{2\left( {{w(f)}^{H}\left( {{M({fT})}{p(f)}} \right)} \right){f}}\end{Bmatrix}} & (35)\end{matrix}$

with the constraints (33b) and (33c) being imposed not on the innerminimization but only on the outer minimization. Since the innerminimization reduces to finding the linear minimum mean-squared error(LMMSE) receiver without any constraint, the solution can be easilyfound as

$\begin{matrix}{{{w_{l\; m\; m\; e}(f)} = \frac{{M({fT})}{R_{N}(f)}^{- 1}{H(f)}{s(f)}}{1 + {\frac{M({fT})}{T}{s(f)}^{H}{(f)}{s(f)}}}},{\forall{f \in}},} & (36)\end{matrix}$

where H(ƒ) is the N(ƒ)-by-N(ƒ) diagonal matrix defined as

H(ƒ)

diag(h(ƒ)′)   (37)

and C(ƒ) is the N(ƒ)-by-N(ƒ) matrix-valued frequency function given by

C(ƒ)

H(ƒ)^(H)R_(N)(ƒ)⁻¹H(ƒ)   (38)

With this solution (36), the MSE performance is given by

$\begin{matrix}{{{{{TM}({fT})}{f}} + {\frac{{TM}({fT})}{1 + {\frac{M({fT})}{T}{s(f)}^{H}{C(f)}{s(f)}}}{f}}},} & (39)\end{matrix}$

because w_(lmmse)(ƒ)=0, ∀ ƒ ∈

.

Now, the outer minimization problem reduces to finding s(ƒ) thatminimizes the MSE in (39) subject to the constraints (33b) and (33c). Toproceed, we define the energy density function a(ƒ) of s(ƒ) as

a(ƒ)

s(ƒ)^(H)s(ƒ)   (40)

for ƒ ∈

. Then, the constraint set can be partitioned into subsets, each ofwhich contains all feasible s(ƒ) having the same energy density functiona(ƒ). So, the outer minimization problem can be rewritten in anequivalent double minimization form

Problem 4:

$\begin{matrix}{\underset{a{(f)}}{minimize}\left\{ {{\begin{matrix}\underset{s{(f)}}{minimize} & {\frac{{TM}({fT})}{1 + {\frac{M({fT})}{T}{s(f)}^{H}{C(f)}{s(f)}}}{f}} \\{{subject}\mspace{14mu} {to}} & {{{{G(f)}^{H}{s(f)}} = 0},{\forall{f \in}}} \\\; & {{{{s(f)}^{H}{s(f)}} = {a(f)}},{\forall{f \in}}}\end{matrix}{subject}\mspace{14mu} {to}\mspace{14mu} \frac{M({fT})}{T}{a(f)}{f}} = A} \right.} & (41)\end{matrix}$

This shows again that an arbitrary choice of the optimal transmitwaveform s_(opt)(ƒ) will do for ƒ ∈

.

To minimize the objective function of the inner optimization problem ofProblem 4, we need to find s_(opt)(ƒ) that maximizes the denominator ofthe integrand at each ƒ ∈

. Hence, the inner optimization problem to find s_(opt)(ƒ) given a(ƒ)snow reduces to a convex maximization problem

Sub-Problem 1:

$\begin{matrix}{\underset{s{(f)}}{maximize}\mspace{14mu} {s(f)}^{H}{C(f)}{s(f)}} & \left( {42a} \right) \\{{{{subject}\mspace{14mu} {to}\mspace{14mu} {G(f)}^{H}{s(f)}} = 0},} & \left( {42b} \right) \\{{{{s(f)}^{H}{s(f)}} = {a(f)}},} & \left( {42c} \right)\end{matrix}$

which needs to be solved for each ƒ ∈

.

Proposition 1: The optimal solution s_(opt)(ƒ) to Sub-Problem 1 is givenby

s _(opt)(ƒ)=√{square root over (a(ƒ))}v _(max)(ƒ)e ^(jθ(ƒ))   (43)

for ƒ ∈

, where θ(ƒ) can be chosen arbitrarily and v_(max)(ƒ) is the normalizedeigenvector corresponding to the largest eigenvalue λ_(max)(ƒ) ofP(ƒ)C(ƒ) or, equivalently, P(ƒ)C(ƒ)P(ƒ) with P(ƒ) being defined as theprojection matrix

$\begin{matrix}\begin{matrix}{{P(f)}\overset{\Delta}{=}{{I(f)} - {{G(f)}\left( {{G(f)}^{H}{G(f)}} \right)^{\dagger}{G(f)}^{H}}}} \\{= {{I(f)} - {{G(f)}{G(f)}^{\dagger}}}}\end{matrix} & (44)\end{matrix}$

where † denotes the Moore-Penrose generalized inverse.

Note that every eigenvector of P(ƒ)C(ƒ) is an eigenvector ofP(ƒ)C(ƒ)P(ƒ) with the same eigenvalue and vice versa, becausepre-multiplying P(ƒ) to P(ƒ)C(ƒ)v(ƒ)=λ(ƒ)v(ƒ) leads to P(ƒ)v(ƒ)=v(ƒ),which means P(ƒ)C(ƒ)P(ƒ)v(ƒ)=λ(ƒ)v(ƒ), and pre-multiplying P(ƒ) toP(ƒ)C(ƒ)P(ƒ)v′(ƒ)=λ′(ƒ)v′(ƒ) leads to P(ƒ)v′(ƒ)=v′(ƒ), which meansP(ƒ)C(ƒ)v′(ƒ)=λ′(ƒ)v′(ƒ).

Using the fact that P(ƒ)v_(max)(ƒ)=v_(max)(ƒ) and the solution (43), theoptimal objective function value of Sub-Problem 1 can be calculated as

$\begin{matrix}{{{s_{opt}(f)}^{H}{C(f)}{s_{opt}(f)}} = {{a(f)}{v_{\max}(f)}^{H}{C(f)}{v_{\max}(f)}}} & \left( {45a} \right) \\{\mspace{211mu} {= {{a(f)}{v_{\max}(f)}^{H}{P(f)}{C(f)}{v_{\max}(f)}}}} & \left( {45b} \right) \\{\mspace{214mu} {= {{a(f)}{\lambda_{\max}(f)}}}} & \left( {45c} \right)\end{matrix}$

for ƒ ∈

. Thus, the outer minimization of Problem 4 to find the optimal energyallotment a_(opt)(ƒ) for ƒ ∈

becomes

Sub-Problem 2

$\begin{matrix}{{\underset{a{(f)}}{minimize}\frac{{TM}({fT})}{1 + {\frac{M({fT})}{T}{\lambda_{\max}(f)}{a(f)}}}{f}}{{{subject}\mspace{14mu} {to}\frac{M({fT})}{T}{a(f)}{f}} = {A.}}} & (46)\end{matrix}$

To find the optimal energy allotment a_(opt)(ƒ) for ƒ ∈

in Sub-Problem 2,

is now replaced by

and the orthogonality constraint (33b) has changed λ_(max)(ƒ) from thelargest eigenvalue of C(ƒ) to the largest eigenvalue of P(ƒ)C(ƒ), sothat λ_(max)(ƒ) can be zero for some ƒ. To proceed, we define the set

_(λ) _(max)

{ƒ ∈

: λ_(max)(ƒ)≠0},   (47)

as the support of λ_(max)(ƒ).

Theorem 1: The VFTs s_(opt)(ƒ) and w_(opt)(ƒ) of the jointly optimaltransmit waveform s_(opt)(t) and the receive waveform w_(opt)(t) as thesolutions to Problem 3 are given, respectively, by

$\begin{matrix}{{s_{opt}(f)} = \left\{ {\begin{matrix}{{\sqrt{a_{opt}(f)}{v_{\max}(f)}^{{j\theta}{(f)}}},} & {\forall{f \in}} \\{{arbitrary},} & {{\forall{f \in}},}\end{matrix}{and}} \right.} & (48) \\{{{w_{opt}(f)} = \frac{{M({fT})}{R_{N}(f)}^{- 1}{H(f)}{s_{opt}(f)}}{1 + {\frac{M({fT})}{T}{s_{opt}(f)}^{H}{C(f)}{s_{opt}(f)}}}},{\forall{f \in}}} & (49)\end{matrix}$

where the optimal energy allotment a_(opt)(ƒ) is given by

a opt  ( f ) = [ v ~ opt - T M  ( fT )  λ max  ( f ) ] +  T M  (fT )  λ max  ( f ) ,  for   f ∈ ⋂ λ  max ( 50 )

with {tilde over (v)}_(opt)>0 being the unique solution to

$\begin{matrix}{{{\left\lbrack {{\overset{\sim}{v}}_{opt} - \sqrt{\frac{T}{{M({fT})}{\lambda_{\max}(f)}}}} \right\rbrack}^{+}\sqrt{\frac{M({fT})}{T\; {\lambda_{\max}(f)}}}{f}} = A} & (51)\end{matrix}$

and [x]+

max (x,0) denoting the positive part of x, and a_(opt)(ƒ)=0 for

∩

.

The solution found in Theorem 1 leads to the jointly minimized MSEε_(opt) given by

$\begin{matrix}{ɛ_{opt} = {{{{TM}({fT})}{f}} + {\frac{{TM}({fT})}{1 + {\frac{M({fT})}{T}{\lambda_{\max}(f)}{a_{opt}(f)}}}{{f}.}}}} & (52)\end{matrix}$

This shows that the minimized MSE is a monotone non-increasing functionof the average transmit power A>0, which justifies the use of theequality constraint P=A instead of the inequality constraint P≦A in theformulation of Problem 1.

B. A Necessary and Sufficient Condition for Existence of Solution

The final question to be answered is what is the necessary andsufficient condition for the existence of a non-trivial optimal solutionsuch that ε_(opt)<

{|b[l]|²}=

TM(ƒT)dƒ. In this subsection, we show that the existence of anon-trivial optimal solution can be determined solely by theorthogonality constraint (33b). To proceed, we prove the followinglemma. Note that the result is not trivial because C(ƒ) is positivesemi-definite in general.

Lemma 2:

P(ƒ)=0

P(ƒ)C(ƒ)P(ƒ)=0.   (53)

Theorem 2: A non-trivial optimal solution exists if the length of theset

{ƒ ∈

: Dim(null space of G(ƒ)^(H))>0}  (54)

is greater than zero, where Dim(•) denotes the dimension of a subspace.

The above result provides a simple way to perform admission controlwithout eigen-decomposing the matrix P(ƒ)C(ƒ) but just using the matrixG(ƒ) that can be obtained once the frequency responses and the samplingtimings of the primary and the virtual primary receivers are identified.

IV. Numerical Results

In this section, we provide numerical results that demonstrate theeffectiveness of our solution in designing cognitive radios. TheMonte-Carlo simulation results as well as the average BER estimatesbased on the conditional Gaussian approximation of the overallinterference are provided. Throughout this section, the primary and thesecondary users employ linear modulation with QPSK symbols, the primarytransmitters have square-root raised cosine pulses as the transmitwaveforms with γ ∈ [0,1] denoting the roll-off factor, and the primaryreceivers employ matched filters matched to the transmit waveform withoutput sampling rate equal to the symbol rate. All the sampling offsetsare set to zero, all the channels are assumed frequency flat, and allthe users have the same E_(b)/N₀.

The first results are to compare the performance of the proposedcognitive radio with that of the cognitive radio utilizing white spaceonly. As illustrated in FIG. 5, there is one primary user that employslinear modulation having the symbol rate of 1/T [Hz] and a square-rootraised cosine (SRRC) pulse as the transmit waveform. It is assumed thatthe primary and the secondary users have the identical symbol rate of1/T [Hz], and that they both employ QPSK symbols with E_(b)/N_(o)=10[dB]. It is also assumed that all the channels are ideal frequency flatchannels with independent and identically distributed (i.i.d.) uniformphase and delay, and that the primary receiver employs a matched filtermatched to the transmit waveform and sampled at the symbol rate. Thetransmit and the receive bands are chosen to be

=

=[−1/T,1/T] [Hz]. The PSDs of the secondary user are also shown in FIG.5, where the length of the spectrum hole is 1/2T or, equivalently,β=0.5, with β ∈ [0,1] being the roll-off factor of the SRRC waveform ofthe primary transmitter. As the relative delays among the channels vary,the optimal waveform of the primary transmitter changes and,consequently, the PSD of the secondary-user signal also changes.

FIG. 6 shows the average bit error rate (BER) performance of thesecondary user versus the length of the spectrum hole, which equals(1−β)/T. It can be seen that the proposed cognitive radio, enjoying theincreased bandwidth, performs significantly better than the cognitiveradio that utilizes the spectrum hole only. Note that the average BERsconverge to the AWGN bound as the length of the spectrum hole increasesto 1/T, the Nyquist minimum bandwidth for zero intersymbol interference.

The next results are to show the trade-off between the average BER andthe symbol rate of the secondary user, especially when there is nospectrum hole. As shown in FIG. 7, there is one primary user with thesymbol rate of 1/T₀ [Hz] and E_(b)/N₀=10 [dB]. The symbol rate 1/T ofthe secondary user is a fraction of 1/T₀, and

=

=[−(1+γ)/(2T₀), (1+γ)/(2T₀)] [Hz]. In particular, the application of theproposed cognitive radio to satellite communication is considered, wherethe primary and the secondary transmitters are terrestrial stations, sothat both the primary- and the secondary-user signals are amplified andretransmitted by a single relay, the satellite. For simplicity, possiblenonlinearity of the satellite amplifier is ignored. Since the cases withT≠T₀ are also considered, the frequency is normalized with respect to1/T₀. As the channels are assumed flat and the propagation delays fromthe satellite to the primary Rx and to the secondary Rx are identical,it turns out that the optimal solution not only makes the secondary-usersignal orthogonal to the primary Rx but also makes the secondary Rx tocompletely null out the primary-user signal. Thus, as shown in FIG. 8,the proposed scheme makes any target BER that is no less than the AWGNbound achievable by trading off the symbol transmission rate, wheneverthe symbol rate 1/T of the secondary user is chosen to satisfy

$\frac{T_{O}}{T} \leq {\gamma.}$

The final results are to compare the performance of the proposedcognitive radio with that of the cognitive radio utilizing white spaceonly. As illustrated in FIG. 9, there are four primary users with γ=0.2,and the length of each guard band is

$\frac{0.2}{T}.$

It is assumed that the channels have independent and identicallydistributed uniform phase and delay. The receive band is

${B_{R} = \left\lbrack \; {{- \frac{3}{T}},\frac{3}{T}} \right\rbrack},$

and the transmit band is

${B_{T} = \left\lbrack {{- \frac{1.2}{T}},\frac{1.2}{T}} \right\rbrack},\left\lbrack {{- \frac{1.4}{T}},\frac{1.4}{T}} \right\rbrack,{\quad{\left\lbrack {{- \frac{1.45}{T}},\frac{1.45}{T}} \right\rbrack,\left\lbrack {{- \frac{1.5}{T}},\frac{1.5}{T}} \right\rbrack,\left\lbrack {{- \frac{2}{T}},\frac{2}{T}} \right\rbrack,\left\lbrack {{- \frac{2.5}{T}},\frac{2.5}{T}} \right\rbrack,{{or}\mspace{14mu}\left\lbrack {{- \frac{3}{T}},\frac{3}{T}} \right\rbrack},}}$

so that the length of the corresponding spectrum hole is0.2/T,0.4/T,0.5/T,0.6/T,0.6/T,0.6/T, or 1.2/T, respectively. To obtainthe optimal solution for each transmit and receive band pair, a propernumber of virtual primary receivers is introduced. FIG. 9 also shows thePSDs of the secondary-user signals when E_(b)/N₀=10 [dB],

=[−1.5/T,1.5/T], the symbol timing offset between two primarytransmitters in the middle is 0.4T, and all the channel delays are zero.As shown in FIG. 10, the proposed cognitive radio, enjoying theincreased bandwidth, performs significantly better than the cognitiveradio that utilizes the spectrum hole only. Note that, due to thecapability of the proposed scheme of utilizing some black and grayspaces, the BER performance close to the AWGN bound is easily achievedeven when the length of the spectrum hole is less than 1/T, the Nyquistminimum bandwidth for zero intersymbol interference.

FIG. 11 shows a method of sharing spectrum with a legacy communicationsystem according an embodiment of the present invention. The method maybe performed by a transmitter of an overlay system.

Referring to FIG. 11, in S700, spectrum correlation of at least onelegacy communication system is acquired. The spectrum correlation may berelated to one of statistical characteristics of the legacycommunication system and/or correlations of signals in the legacycommunication system. The spectrum correlation is used to determine2-dimensional power spectrum density (PSD) of the legacy communicationsystem and is related to correlations of signals of the legacycommunication system in the frequency domain. The spectrum correlationis used to determine PSD R_(N)(ƒ) as shown in Equation (19) and isrepresented as

{N(t)N(s)*}.

In S750, a transmit waveform is generated based on the spectrumcorrelation. A transmit signal is generated by combining the transmitwaveform and a data signal and is transmitted through a radio channel.The transmit waveform may be generated by using Vectorized FourierTransform (VFT).

The transmit waveform can directly be generated from the spectrumcorrelation. The transmitter may acquire the spectrum correlation frominformation transmitted from the legacy communication system.Alternatively, the transmitter may acquire the spectrum correlation bysensing the signals of the legacy communication system.

FIG. 12 shows a flowchart of generating a transmit waveform. In S810, aPSD matrix R_(N)(ƒ) is determined based on the spectrum correlation asshown in Equation (19). In S820, a channel correlation matrix C(ƒ) byusing the PSD matrix R_(N)(ƒ) and a channel matrix H(ƒ) is determined asshown in Equation (38). In S830, a projection matrix P(ƒ) is determinedas shown in Equation (44). The projection matrix P(ƒ) may be determinedby using a blocking matrix G(ƒ) obtained from impulse responses of thelegacy communication system. In S840, an eigenvalue λ_(max)(ƒ) ofP(ƒ)C(ƒ) is determined. The eigenvalue λ_(max)(ƒ) may be the largesteigenvalue of the P(ƒ)C(ƒ). In S850, a normalized eigenvector v_(max)(ƒ)corresponding to the λ_(max)(ƒ) is determined. In S860, a transmitwaveform s_(opt)(ƒ) from the eigenvector v_(max)(ƒ) is determined asshown in Equation (48).

FIG. 13 shows a block diagram of a transmitter and a receiver of anoverlay communication system according an embodiment of the presentinvention. A transmitter 800 includes a data processor 810, a waveformgenerator 850 and a Radio Frequency (RF) unit 880. The data processor810 is configured to generate a data signal. The waveform generator 850is configured to generate a transmit waveform based on spectrumcorrelation of the legacy communication system. The transmit waveformmay be generated by using Vectorized Fourier Transform (VFT). The RFunit 880 is configured to transmit a radio signal based on the transmitwaveform. The radio signal is generated by combining the data signal andthe transmit waveform. The radio signal is transmitted through atransmit antenna 890.

A receiver 900 includes a data processor 910, a waveform generator 950and a RF filter 980. A receive signal which is transmitted by thetransmitter 800 is received through a receive antenna 990. The RF filter980 is configured to filter the receive signal based on a receivewaveform. The waveform generator 950 is configured to generate thereceive waveform based on spectrum correlation of the legacycommunication system. The data processor 910 reproduces data from thefiltered signal. The receive waveform w_(opt)(ƒ) as shown in Equation(49) may be acquired through steps S810 to S850 in FIG. 12.

Spectrum correlation of one or more legacy communication systems may begiven in various ways. Spectrum correlation may be information on PSD ofthe legacy communication system or information on parameter to determinethe PSD of the legacy communication system.

Unlike conventional cognitive radios that utilize spectrum holes only,the proposed method can also utilize spectrally correlated frequencycomponents where primary-user signals are present. Under fixed linearreceiver assumption on the primary users, the optimal transmit andreceive waveforms of the secondary user are derived that jointlyminimize the MSE at the output of the linear receiver of the secondaryuser, subject to zero interference at the output samples of the primaryreceivers. Using the VFT technique, the optimal solution and thenecessary and sufficient condition for the existence of the optimalsolution are derived in the frequency domain. It is shown that theproposed scheme significantly improves the BER performance and thespectral efficiency without modifying the primary-user side.

All the functions described above may be performed by a processor suchas a microprocessor, controller, microcontroller, application specificintegrated circuit (ASIC), and the like operated based on software, aprogram code, or the like coded to perform the functions. A design,development, and implementation of the code will be apparent to thoseskilled in the art based on the description of the present invention.

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the spirit or scope of the invention. Thus, it isintended that the present invention cover the modifications andvariations of this invention provided they come within the scope of theappended claims and their equivalents.

1. A method of sharing spectrum with a legacy communication system, themethod comprising: acquiring spectrum correlation of the legacycommunication system; and generating a transmit waveform based on thespectrum correlation.
 2. The method of claim 1, further comprising:generating a transmit signal by combining the transmit waveform and adata signal; and transmitting the transmit signal.
 3. The method ofclaim 1, wherein the spectrum correlation is related to statisticalcharacteristics of the legacy communication system.
 4. The method ofclaim 1, wherein the spectrum correlation is related to correlations ofsignals of the legacy communication system in frequency domain.
 5. Themethod of claim 1, wherein the transmit waveform is generated by usingVectorized Fourier Transform (VFT).
 6. The method of claim 1, whereingenerating the transmit waveform comprises: determining a power spectraldensity (PSD) matrix R_(N)(ƒ) based on the spectrum correlation;determining a channel correlation matrix C(ƒ) by using the PSD matrixR_(N)(ƒ) and a channel matrix H(ƒ); determining a projection matrixP(ƒ); determining an eigenvalue λ_(max)(ƒ) of P(ƒ)C(ƒ); determining anormalized eigenvector v_(max)(ƒ) corresponding to the λ_(max)(ƒ); anddetermining the transmit waveform s_(opt)(ƒ) from the eigenvectorv_(max)(ƒ).
 7. The method of claim 6, wherein the eigenvalue λ_(max)(ƒ)is the largest eigenvalue of the P(ƒ)C(ƒ).
 8. The method of claim 6,wherein the projection matrix P(ƒ) is determined by using a blockingmatrix G(ƒ) obtained from impulse responses of the legacy communicationsystem.
 9. The method of claim 1, wherein the spectrum correlation isobtained from the received signal from the legacy communication systems.10. A transmitter for sharing spectrum with a legacy communicationsystem, comprising: a waveform generator for generating a transmitwaveform; and a Radio Frequency (RF) unit for transmitting a radiosignal based on the transmit waveform, wherein the waveform generator isconfigured to: acquire spectrum correlation of the legacy communicationsystem; and generate the transmit waveform based on the spectrumcorrelation.
 11. The transmitter of claim 10, wherein the spectrumcorrelation is received from the legacy communication system.
 12. Thetransmitter of claim 10, further comprising: a data processor forgenerating a data signal, wherein the radio signal is generated bycombining the data signal and the transmit waveform.
 13. The transmitterof claim 10, wherein the transmit waveform is generated by usingVectorized Fourier Transform (VFT).
 14. A receiver for sharing spectrumwith a legacy communication system, comprising: a waveform generator forgenerating a receive waveform; and a Radio Frequency (RF) filter forfiltering a receive signal based on the receive waveform, wherein thewaveform generator is configured to: acquire spectrum correlation of thelegacy communication system; and generate the receive waveform based onthe spectrum correlation.
 15. The receiver of claim 14, wherein thewaveform generator is configured to: determine a power spectral density(PSD) matrix R_(N)(ƒ) based on the spectrum correlation; determine achannel correlation matrix C(ƒ) by using the PSD matrix R_(N)(ƒ) and achannel matrix H(ƒ); determine a projection matrix P(ƒ); determine aneigenvalue λ_(max)(ƒ) of P(ƒ)C(ƒ); determine a normalized eigenvectorv_(max)(ƒ) corresponding to the λ_(max)(ƒ); and determine the receivewaveform w_(opt)(ƒ) from the eigenvector v_(max)(ƒ).
 16. The receiver ofclaim 15, wherein the eigenvalue λ_(max)(ƒ) is the largest eigenvalue ofthe P(ƒ)C(ƒ).
 17. The receiver of claim 14, wherein the spectrumcorrelation is received from the legacy communication system.
 18. Thereceiver of claim 14, wherein the spectrum correlation is received froma transmitter which transmits the receive signal.